Honors Physics “Mechanics for Physicists and Engineers” Agenda for Today

1. Honors Physics “Mechanics for Physicists and Engineers”Agenda for Today • Advice • 1-D Kinematics • Average & instantaneous velocity and acceleration • Motion with constant acceleration • Freefall

2. Kinematics Objectives • Define average and instantaneous velocity • Caluclate kinematic quantities using equations • interpret and plot position -time graphs • be able to determine and describe the meaning of the slope of a position-time graph

3. Kinematics • Location and motion of objects is described using Kinematic Variables: • Some examples of kinematic variables. • position r vector, (d,x,y,z) • velocity v vector • acceleration a vector • Kinematic Variables: • Measured with respect to a reference frame. (x-y axis) • Measured using coordinates (having units). • Many kinematic variables are Vectors, which means they have a direction as well as a magnitude. • Vectors denoted by boldface Vor arrow above the variable

4. Motion • Position: Separation between an object and a reference point (Just a point) • Distance: Separation between two objects • Displacement of an object is the distance between it’s final position df and it’s initial position d i (d f - di)= d • Scalar: Quantity that can be described by a magnitude(strength) only • Distance, temperature, pressure etc.. • Vector: A quantity that can be described by both a magnitude and direction • Force, displacement, torque etc.

5. Speed and Velocity • Speed describes the rate at which an object moves. Distance traveled per unit of time. • Velocity describes an objects’ speed and direction. • Approximate units of speed

6. Motion in 1 dimension • In general, position at time t1 is usually denoted d, r(t1) or x(t1) • In 1-D, we usually write position as x(t1 ) but for this level we’ll use d • Since it’s in 1-D, all we need to indicate direction is + or . • Displacement in a time t = t2 - t1isx = x2 - x1= d2 -d1 x some particle’s trajectoryin 1-D x2 x x1 t1 t2 t t

7. 1-D kinematics • Velocity v is the “rate of change of position” • Average velocity vav in the time t = t2 - t1is: x trajectory d2 x Vav = slope of line connecting x1 and x2. d1 t1 t2 t t

8. 1-D kinematics... • Instantaneous velocity v is definedas the velocity at an instant of time (t= 0) • Slope formula becomes undefined at t = 0 x soV(t2 ) = slope of line tangent to path at t2. x2 x x1 Calculus Notation t1 t2 t t

9. v 60 t 1 2 More 1-D kinematics • We saw that v = x / t • so therefore x = v t( i.e. 60 mi/hr x 2 hr = 120 mi ) • See text: 3.2 • In “calculus” language we would write dx = v dt, which we can integrate to obtain: • Graphically, this is adding up lots of small rectangles: v(t) + +...+ = displacement t

10. 1-D kinematics... • Acceleration a is the “rate of change of velocity” • Average acceleration aav in the time t = t2 - t1is: • Andinstantaneous acceleration a is definedas:The acceleration when t = 0 . Same problem as instantaneous velocity. Slope equals line tangent to path of velocity vs time graph.

11. Problem Solving • Read ! • Before you start work on a problem, read the problem statement thoroughly. Make sure you understand what information in given, what is asked for, and the meaning of all the terms used in stating the problem. • Watch your units ! • Always check the units of your answer, and carry the units along with your numbers during the calculation. • Understand the limits ! • Many equations we use are special cases of more general laws. Understanding how they are derived will help you recognize their limitations (for example, constant acceleration).

12. IV. Displacement during acceleration. • You accelerate from 0 m/s to 30 m/s in 3 seconds, how far did you travel? • What if a car initially at 10 m/s, accelerates at a rate of 5 m/s2 for 7 seconds. How far does it move? • df=1/2at2 + vit + di • C. An airplane must reach a speed of 71 m/s for a successful takeoff. What must be the rate of acceleration if the runway is 1.0 km long? • d = (vf2 - vi2) /2a

13. Recap • If the position x is known as a function of time, then we can find both velocity vand acceleration a as a function of time! x t v t a t

14. Recap • So for constant acceleration we find: x t v • From which we can derive: t a t

15. IV. Acceleration due to gravity • The acceleration of a freely falling object is 9.8 m/s2 (32 ft/s2) towards the earth. • The farther away from the earth’s center, the smaller the value of the acceleration due to gravity. For activities near the surface of the earth (within 5-6 km or more) we will assume g=9.8 m/s2 (10 m/s2). • Neglecting air resistance, an object has the same acceleration on the way up as it does on the way down. • Use the same equations of motion but substitute the value of ‘g’ for acceleration ‘a’.

16. Recap of kinematics lectures • Measurement and Units (Chapter 1) • Systems of units • Converting between systems of units • Dimensional Analysis • 1-D Kinematics • Average & instantaneous velocity and and acceleration • Motion with constant acceleration